186 



STRENGTH OF MATERIALS 



From Article 66, the angular deformation dff is given by the expression 



Mds 



Consequently 



and hence the total change in length of the span is 



* 



-fi 



1-1=--* 



Therefore the condition that the span shall be unchanged in length 

 by the strain is 



The bending moment M in tliis expression has the same value as 

 in the preceding article, namely, M = M p - P k z. Inserting this value 



ui J/in the above condi- 

 tion, it becomes 



FIG. 135 



from which, as in the pre- 

 ceding article, 



, 



153. Graphical deter- 

 mination of the linear 



arch. From the condition that the bending stress sliall produce no 

 change in the length of the span, the position of the linear arch may 

 be determined graphically as follows. 



In Fig. 135 let ACF represent the center line of the rib. A />/' tin- 

 corresponding equilibrium polygon drawn to any convenient scale, and 

 ABF the linear arch. Then the linear arch can be obtained from the 

 equilibrium polygon by reducing each ordinate of the latter in a certain 



* The effects of changes of temperature :iml also of direct compn - in altering 



the length of the span are neglected, as they are slight in comparison with that due to 

 bending strain. 



